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<h2> SECT. XI. OF THE PROBABILITY OF CHANCES. </h2>
<p>But in order to bestow on this system its full force and evidence, we must
carry our eye from it a moment to consider its consequences, and explain
from the same principles some other species of reasoning, which are
derived from the same origin.</p>
<p>Those philosophers, who have divided human reason into knowledge and
probability, and have defined the first to be that evidence, which arises
from the comparison of ideas, are obliged to comprehend all our arguments
from causes or effects under the general term of probability. But though
every one be free to use his terms in what sense he pleases; and
accordingly in the precedent part of this discourse, I have followed this
method of expression; it is however certain, that in common discourse we
readily affirm, that many arguments from causation exceed probability, and
may be received as a superior kind of evidence. One would appear
ridiculous, who would say, that it is only probable the sun will rise
to-morrow, or that all men must dye; though it is plain we have no further
assurance of these facts, than what experience affords us. For this
reason, it would perhaps be more convenient, in order at once to preserve
the common signification of words, and mark the several degrees of
evidence, to distinguish human reason into three kinds, viz. THAT FROM
KNOWLEDGE, FROM PROOFS, AND FROM PROBABILITIES. By knowledge, I mean the
assurance arising from the comparison of ideas. By proofs, those
arguments, which are derived from the relation of cause and effect, and
which are entirely free from doubt and uncertainty. By probability, that
evidence, which is still attended with uncertainty. It is this last
species of reasoning, I proceed to examine.</p>
<p>Probability or reasoning from conjecture may be divided into two kinds,
viz. that which is founded on chance, and that which arises from causes.
We shall consider each of these in order.</p>
<p>The idea of cause and effect is derived from experience, which presenting
us with certain objects constantly conjoined with each other, produces
such a habit of surveying them in that relation, that we cannot without a
sensible violence survey them iii any other. On the other hand, as chance
is nothing real in itself, and, properly speaking, is merely the negation
of a cause, its influence on the mind is contrary to that of causation;
and it is essential to it, to leave the imagination perfectly indifferent,
either to consider the existence or non-existence of that object, which is
regarded as contingent. A cause traces the way to our thought, and in a
manner forces us to survey such certain objects, in such certain
relations. Chance can only destroy this determination of the thought, and
leave the mind in its native situation of indifference; in which, upon the
absence of a cause, it is instantly re-instated.</p>
<p>Since therefore an entire indifference is essential to chance, no one
chance can possibly be superior to another, otherwise than as it is
composed of a superior number of equal chances. For if we affirm that one
chance can, after any other manner, be superior to another, we must at the
same time affirm, that there is something, which gives it the superiority,
and determines the event rather to that side than the other: That is, in
other words, we must allow of a cause, and destroy the supposition of
chance; which we had before established. A perfect and total indifference
is essential to chance, and one total indifference can never in itself be
either superior or inferior to another. This truth is not peculiar to my
system, but is acknowledged by every one, that forms calculations
concerning chances.</p>
<p>And here it is remarkable, that though chance and causation be directly
contrary, yet it is impossible for us to conceive this combination of
chances, which is requisite to render one hazard superior to another,
without supposing a mixture of causes among the chances, and a conjunction
of necessity in some particulars, with a total indifference in others.
Where nothing limits the chances, every notion, that the most extravagant
fancy can form, is upon a footing of equality; nor can there be any
circumstance to give one the advantage above another. Thus unless we
allow, that there are some causes to make the dice fall, and preserve
their form in their fall, and lie upon some one of their sides, we can
form no calculation concerning the laws of hazard. But supposing these
causes to operate, and supposing likewise all the rest to be indifferent
and to be determined by chance, it is easy to arrive at a notion of a
superior combination of chances. A dye that has four sides marked with a
certain number of spots, and only two with another, affords us an obvious
and easy instance of this superiority. The mind is here limited by the
causes to such a precise number and quality of the events; and at the same
time is undetermined in its choice of any particular event.</p>
<p>Proceeding then in that reasoning, wherein we have advanced three steps;
that chance is merely the negation of a cause, and produces a total
indifference in the mind; that one negation of a cause and one total
indifference can never be superior or inferior to another; and that there
must always be a mixture of causes among the chances, in order to be the
foundation of any reasoning: We are next to consider what effect a
superior combination of chances can have upon the mind, and after what
manner it influences our judgment and opinion. Here we may repeat all the
same arguments we employed in examining that belief, which arises from
causes; and may prove, after the same manner, that a superior number of
chances produces our assent neither by demonstration nor probability. It
is indeed evident that we can never by the comparison of mere ideas make
any discovery, which can be of consequence in this affairs and that it is
impossible to prove with certainty, that any event must fall on that side
where there is a superior number of chances. To, suppose in this case any
certainty, were to overthrow what we have established concerning the
opposition of chances, and their perfect equality and indifference.</p>
<p>Should it be said, that though in an opposition of chances it is
impossible to determine with certainty, on which side the event will fall,
yet we can pronounce with certainty, that it is more likely and probable,
it will be on that side where there is a superior number of chances, than
where there is an inferior: should this be said, I would ask, what is here
meant by likelihood and probability? The likelihood and probability of
chances is a superior number of equal chances; and consequently when we
say it is likely the event win fall on the side, which is superior, rather
than on the inferior, we do no more than affirm, that where there is a
superior number of chances there is actually a superior, and where there
is an inferior there is an inferior; which are identical propositions, and
of no consequence. The question is, by what means a superior number of
equal chances operates upon the mind, and produces belief or assent; since
it appears, that it is neither by arguments derived from demonstration,
nor from probability.</p>
<p>In order to clear up this difficulty, we shall suppose a person to take a
dye, formed after such a manner as that four of its sides are marked with
one figure, or one number of spots, and two with another; and to put this
dye into the box with an intention of throwing it: It is plain, he must
conclude the one figure to be more probable than the other, and give the
preference to that which is inscribed on the greatest number of sides. He
in a manner believes, that this will lie uppermost; though still with
hesitation and doubt, in proportion to the number of chances, which are
contrary: And according as these contrary chances diminish, and the
superiority encreases on the other side, his belief acquires new degrees
of stability and assurance. This belief arises from an operation of the
mind upon the simple and limited object before us; and therefore its
nature will be the more easily discovered and explained. We have nothing
but one single dye to contemplate, in order to comprehend one of the most
curious operations of the understanding.</p>
<p>This dye, formed as above, contains three circumstances worthy of our
attention. First, Certain causes, such as gravity, solidity, a cubical
figure, &c. which determine it to fall, to preserve its form in its
fall, and to turn up one of its sides. Secondly, A certain number of
sides, which are supposed indifferent. Thirdly, A certain figure inscribed
on each side. These three particulars form the whole nature of the dye, so
far as relates to our present purpose; and consequently are the only
circumstances regarded by the mind in its forming a judgment concerning
the result of such a throw. Let us, therefore, consider gradually and
carefully what must be the influence of these circumstances on the thought
and imagination.</p>
<p>First, We have already observed, that the mind is determined by custom to
pass from any cause to its effect, and that upon the appearance of the
one, it is almost impossible for it not to form an idea of the other.
Their constant conjunction in past instances has produced such a habit in
the mind, that it always conjoins them in its thought, and infers the
existence of the one from that of its usual attendant. When it considers
the dye as no longer supported by the box, it can not without violence
regard it as suspended in the air; but naturally places it on the table,
and views it as turning up one of its sides. This is the effect of the
intermingled causes, which are requisite to our forming any calculation
concerning chances.</p>
<p>Secondly, It is supposed, that though the dye be necessarily determined to
fall, and turn up one of its sides, yet there is nothing to fix the
particular side, but that this is determined entirely by chance. The very
nature and essence of chance is a negation of causes, and the leaving the
mind in a perfect indifference among those events, which are supposed
contingent. When therefore the thought is determined by the causes to
consider the dye as falling and turning up one of its sides, the chances
present all these sides as equal, and make us consider every one of them,
one after another, as alike probable and possible. The imagination passes
from the cause, viz. the throwing of the dye, to the effect, viz. the
turning up one of the six sides; and feels a kind of impossibility both of
stopping short in the way, and of forming any other idea. But as all these
six sides are incompatible, and the dye cannot turn up above one at once,
this principle directs us not to consider all of them at once as lying
uppermost; which we look upon as impossible: Neither does it direct us
with its entire force to any particular side; for in that case this side
would be considered as certain and inevitable; but it directs us to the
whole six sides after such a manner as to divide its force equally among
them. We conclude in general, that some one of them must result from the
throw: We run all of them over in our minds: The determination of the
thought is common to all; but no more of its force falls to the share of
any one, than what is suitable to its proportion with the rest. It is
after this manner the original impulse, and consequently the vivacity of
thought, arising from the causes, is divided and split in pieces by the
intermingled chances.</p>
<p>We have already seen the influence of the two first qualities of the dye,
viz. the causes, and the number and indifference of the sides, and have
learned how they give an impulse to the thought, and divide that impulse
into as many parts as there are unites in the number of sides. We must now
consider the effects of the third particular, viz. the figures inscribed
on each side. It is evident that where several sides have the same figure
inscribe on them, they must concur in their influence on the mind, and
must unite upon one image or idea of a figure all those divided impulses,
that were dispersed over the several sides, upon which that figure is
inscribed. Were the question only what side will be turned up, these are
all perfectly equal, and no one coued ever have any advantage above
another. But as the question is concerning the figure, and as the same
figure is presented by more than one side: it is evident, that the
impulses belonging to all these sides must re-unite in that one figure,
and become stronger and more forcible by the union. Four sides are
supposed in the present case to have the same figure inscribed on them,
and two to have another figure. The impulses of the former are, therefore,
superior to those of the latter. But as the events are contrary, and it is
impossible both these figures can be turned up; the impulses likewise
become contrary, and the inferior destroys the superior, as far as its
strength goes. The vivacity of the idea is always proportionable to the
degrees of the impulse or tendency to the transition; and belief is the
same with the vivacity of the idea, according to the precedent doctrine.</p>
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